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In this paper we study the nonhomogeneous semilinear fractional Schr?dinger equation with critical growth■ where s ∈(0,1),N 4 s,and λ 0 is a parameter,2_s~*=2 N/N-2 s is the fractional critical Sobolev exponent,f and h are some given functions.We show that there exists 0 λ~*+∞such that the problem has exactly two positive solutions if λ∈(0,λ~*),no positive solutions for λλ~*,a unique solution(λ~*,u_(λ~*))if λ=λ~*,which shows that(λ~*,u_(λ~*)) is a turning point in H~s(R~N) for the problem.Our proofs are based on the variational methods and the principle of concentration-compactness.  相似文献   

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This paper is dedicated to studying the following elliptic system of Hamiltonian type:■where N≥3,V,Q∈C(RN,R),V(x)is allowed to be sign-changing and inf Q>0,and F∈C1(R2,R)is superquadratic at both 0 and infinity but subcritical.Instead of the reduction approach used in Ding et al.(2014),we develop a more direct approach—non-Nehari manifold approach to obtain stronger conclusions but under weaker assumptions than those in Ding et al.(2014).We can find anε0>0 which is determined by terms of N,V,Q and F,and then we prove the existence of a ground state solution of Nehari-Pankov type to the coupled system for allε∈(0,ε0].  相似文献   

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In this paper, we study the existence and concentration of positive solution of a class of coupled Schrödinger equations. We admit that the potentials may not be non-negative and suppose that the intersection of the sets has positive Lebesgue measure. By studying the modified functional of the associated functional carefully, we establish the existence of positive least energy solutions for the coupled Schrödinger system. Moreover, we prove the concentration phenomenon of the positive solution when the parameter goes to infinity.  相似文献   

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We prove the existence of periodic solutions and solitons in the nonlinear Schrödinger equation with a nonlocal integral term of convolution type. By separating phase and amplitude, the problem is reduced to an integro-differential formulation that can be written as a fixed point problem for a suitable operator on a Banach space. Then a fixed point theorem due to Krasnoselskii can be applied.  相似文献   

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We prove the existence of positive solutions with optimal local regularity of the homogeneous equation of Schr?dinger type $$ - {\rm{div}}(A\nabla u) - \sigma u = 0{\rm{ in }}\Omega $$ for an arbitrary open ?? ? ? n under only a form-boundedness assumption on ?? ?? D??(??) and ellipticity assumption on A ?? L ??(??) n×n . We demonstrate that there is a two-way correspondence between form boundedness and existence of positive solutions of this equation as well as weak solutions of the equation with quadratic nonlinearity in the gradient $$ - {\rm{div}}(A\nabla u) = (A\nabla v) \cdot \nabla v + \sigma {\rm{ in }}\Omega $$ As a consequence, we obtain necessary and sufficient conditions for both formboundedness (with a sharp upper form bound) and positivity of the quadratic form of the Schr?dinger type operator H = ?div(A?·)-?? with arbitrary distributional potential ?? ?? D??(??), and give examples clarifying the relationship between these two properties.  相似文献   

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In this paper we consider the following elliptic system in \mathbbR3{\mathbb{R}^3}
$\qquad\left\{{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\right.$\qquad\left\{\begin{array}{ll}-\Delta u+u+\lambda K(x)\phi u=a(x)|u|^{p-1}u \quad &x \in {\mathbb{R}}^{3}\\ -\Delta \phi=K(x)u^{2} \quad &x \in {\mathbb{R}}^{3}\end{array}\right.  相似文献   

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In this paper,we first establish narrow region principle and decay at infinity theorems to extend the direct method of moving planes for general fractional p-Laplacian systems.By virtue of this method,we investigate the qualitative properties of positive solutions for the following Schrodinger system with fractional p-Laplacian{(-△)spu+aup-1=f(u,v),(-△)tpv+bv(p-1)=g(u,v),where 0N(N≥2),the monotonicity in the parabolic domain and the nonexistence on the half space for positive solutions to the above system under some suitable conditions on f and g,respectively.  相似文献   

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